Homology Sheaves and Cohomology Groups

Abstract

The injective model structure for simplicial presheaves induces a model structure on the categories of presheaves of simplicial abelian groups and chain complexes, defined so that the underlying simplicial presheaf functor and the free abelian functor form a Quillen adjunction. This is the setting for the homotopy theoretic approach to sheaf cohomology theory and hypercohomology, and their classifications as morphisms in the homotopy category of simplicial presheaves.

These model structures can be localized, by the method introduced in Chapter 7. Examples include the motivic model structure for presheaves of chain complexes on the Nisnevich site of a scheme. The localization technique can be extended to additive contravariant functors to chain complexes which are defined on categories with objects in a site which are enriched in abelian groups. Sample outcomes include Voevodsky's category of effective motives on a perfect field.